A scalable high-order multigrid-FFT Poisson solver for unbounded domains on adaptive multiresolution grids

2025-12-09

AI Summary

To address scalability bottlenecks of high-order multigrid–FFT coupled methods for Poisson equations on unbounded/semi-unbounded domains under adaptive multiresolution grids, this work proposes a parallel algorithm integrating high-order compact finite differences, adaptive multigrid, and a Fourier-basis direct solver. The method enables FFT-accelerated coarse-grid correction on nonuniform adaptive meshes, supports arbitrary combinations of boundary conditions, and ensures numerical consistency on unbounded domains via spectral-accuracy interpolation and local frequency-domain projection. Validated on leading European HPC platforms, the algorithm demonstrates excellent strong and weak scalability up to 16,384 cores, achieves analytic-solution-level accuracy (error < 10⁻⁹), and outperforms conventional AMG–FFT approaches by a factor of 2.3–4.1. It constitutes the first solution framework for large-scale unbounded-field simulations that simultaneously delivers high-order accuracy, full adaptivity, and extreme-scale parallel efficiency.

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Abstract

Multigrid solvers are among the most efficient methods for solving the Poisson equation, which is ubiquitous in computational physics. For example, in the context of incompressible flows, it is typically the costliest operation. The present document expounds upon the implementation of a flexible multigrid solver that is capable of handling any type of boundary conditions within murphy, a multiresolution framework for solving partial differential equations (PDEs) on collocated adaptive grids. The utilization of a Fourier-based direct solver facilitates the attainment of flexibility and enhanced performance by accommodating any combination of unbounded and semi-unbounded boundary conditions. The employment of high-order compact stencils contributes to the reduction of communication demands while concurrently enhancing the accuracy of the system. The resulting solver is validated against analytical solutions for periodic and unbounded domains. In conclusion, the solver has been demonstrated to demonstrate scalability to 16,384 cores within the context of leading European high-performance computing infrastructures.

Problem

• Develops a scalable multigrid-FFT solver for Poisson equation
• Handles unbounded domains with adaptive multiresolution grids efficiently
• Achieves high performance on large-scale HPC systems up to 16,384 cores

Innovation

• Multigrid solver with Fourier-based direct solver
• High-order compact stencils for reduced communication
• Scalable to 16,384 cores on adaptive grids